No! It is assumed that any car you drive when you buy a Non-Owner Policy in Oregon DOES NOT belong to you. In other words, it doesn't make sense for you to purchase a Non-Owner policy if you own a car. Owning a car is defined as having a vehicle registered in your name in the database of the Oregon DMV or any other state, for that matter. So, we presume that if you buy a Non-Owner policy, you will be driving vehicles that belong to someone else.
When would you be driving a car that belongs to someone else? Often, your employer may ask you to drive a company-owned automobile. The company vehicle must have insurance provided by the company. You should ask to see where the proof of insurance and registration are before you begin to operate the car. The owner of the company should exercise caution before allowing you to drive the vehicle, as most commercial insurance companies require each driver to be listed on the company policy before they are authorized to operate it.
Another possibility is that you have a friend or family member who will lend you a vehicle. BE SURE IT HAS INSURANCE ON IT BY ITS OWNER. Your Non-Owner Policy IS NOT a replacement for the insurance the owner of the car should have purchased. So, if your friend or family member offers you a car that is not insured, you should decline to use it. The vehicle could be impounded and taken by a tow truck if you are pulled over by the police, even if you show proof you have purchased a Non-Owner Policy! And you could still be cited for driving without insurance, which, if convicted, could cause your three-year SR22 requirement to start all over.